Why do we use numerical integration in modelling?
Numerical integration is the approximation of a continuous model in discrete steps (both in time and space for spatial models)
What are the advantages of using numerical integration for modelling instead of analytical solutions?
- Many models are not solvable analytically, but always numerically
- With numerical integration, there is virtually no limit to the complexity of models that can be solved:
* Driving variables (e.g. time series of weather or deposition)
* Discrete events (e.g. harvesting)
Equation for logistic growth rate?
dN/dt = r * N * (1 - N/K)
K = carrying capacity
What is a driving variable?
Driving variables characterize the influence of external factors on the system and may thus be essential to the model, yet they are not influenced by the processes within the system and thus not explicitly modelled.
What do Ordinary Differential Equations describe?
The change in states
How can be ODEs mathematically represented?
Y’ = f(Y, p, t)
What is a condition to solve this equation Y’ = f(Y, p, t)?
The initial state needs to be known (state Y at time = 0, Y_0)
What is the mathematical representation of the forward finite difference method?
Y_t+dt = Y_t + dt * f(Y, p, t)
In a logistic growth model describe what happens when N (state) is very small
dN/dt = r * N * (1 - N/K)
if N= very small (0) then eq becomes
dN/dt = r * N * 1
meaning that the growth rate is now exponential
In a logistic growth model describe what happens when N (state) is equal or approaching K
dN/dt = r * N * (1 - N/K)
if N= K then eq becomes
dN/dt = r * N * (1 - K/K)
= r * N * (1 - 1) =
= r * N * 0 =
= 0
meaning that the growth rate is now 0 thus the population stops growing
In a logistic growth model describe what happens when N (state) is bigger than K
dN/dt = r * N * (1 - N/K)
if N > K then eq becomes
dN/dt = r * N * (1 - 2)
= r * N * - (value) –> whole eq become negative, thus the growth rate will also be negative
What is the analytical solution to a simple logistic model?
N_t = (K * N_0 * e ^ (r *t)) / (K + N_0 * (e ^ (r * t) - 1))
What are the components used in the ode() function in R?
- the value(s) of initial state variable(s) – Y0
- the value(s) of parameter(s) – p
- a function with the auxiliary and differential equations returning the rates of all state variables –
f - time – t
When is the highest growth in a logistical equation?
when N = K/2
What happens when the time step we chose for the model is too large?
The model oscillates around the equilibrium
What is the order of variables when inputting in function ode()?
ode(y = state
t = time sequence
func = logistic growth function
parms = parameters
method = euler’s/runge-kutta)
What should event data frames contain?
- var: the state variable name (or number) that is affected by the event;
- time: the time at which the event is to take place;
- value: the value, magnitude of the event;
- method: which event is to take place; it should be one of “replace”, “add”, “multiply” (but it is also allowed to specify the number 1 = replace, 2 = add, 3 = multiply)
How are driving variables included in the ode() function?
They are included through forcing functions.
What are forcing functions?
Functions that appear in the equations and are only a function of time, and not of any of the other variables.
When dealing with driving variables in a model, how do we usually have data and what should we do to make it continuous?
Often, we have data on driving variables gathered as a time series, containing values only at specified times.
Thus, if a model uses driving variables, their value at each time point needs to be estimated by interpolation of the data series.
How does the 4th-order Runge-Kutta differ from Euler’s method of integration?
4th-order Runge-Kutta is a weighted form of Euler’s with different rates of change. RK4 involves four function evaluations per step. The different rates of change are calculated from the initial starting point. First, the rate of change is calculated for half a time step and the final and fourth one is calculated for the whole time step. Makes the approximation better.
…
How do we measure the approximation of the model?
- Accuracy (small numerical error)
- Efficiency (fast to solve)
- Stability (no oscillations or chaotic behaviour)
- Memory usage
How to make the approximation better?
- Make Δt smaller
- Evaluation at multiple points in time during ∆t
- Including higher order derivatives e.g. using a Taylor expansion
- Using a variable time-step instead of a fixed time-step
- Small time step when dynamics are fast
- Large time step when dynamics are slow
Euler’s and Runge-Kutta mathematical equations
- Euler’s
y_t+dt = y_t + dt * dy/dt - Runge-kutta
y_t + dt = y_t + 1/6 * dt * (k_1 + 2k_2 + 2k_3 + k_4)
